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Quantum Computation with coupled quantum dots
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• Two sides of a coin
• Two different polarization of a photon
• Alignment of a nuclear spin in a uniform magnetic
  field
• Two states of an electron orbiting a single atom
• .
• .
• .

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CNOT single qubit gates form a universal
Set. Quantum Algorithm is a (2N)x(2N)
unitary
Qubit Manipulation Gate qubit
operation
Single qubit gate Hadamard gate H Pauli gates
X, Y and Z Two qubit gate Swap gate
Controlled-Not gate or CNOT,
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Physical systems actively considered for quantum
computer implementation
Electrons on liquid He Small Josephson
junctions charge qubits flux qubits
Spin spectroscopies, impurities in
semiconductors Coupled quantum dots
Qubits spin,charge,excitons Exchange coupled,
cavity coupled
Liquid-state NMR NMR spin lattices Linear
ion-trap spectroscopy Neutral-atom
optical lattices Cavity QED atoms Linear
optics with single photons Nitrogen vacancies
in diamond Topologically ordered materials
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DiVincenzo Criteria
A qubit must be well characterized in the sense
that one has a good theoretical description not
only of the qubit itself (in terms of an
internal Hamiltonian, accurate knowledge of all
physical parameters, etc.), but also of all
relevant mechanisms that couple qubits among
each other and to the environment. The word
scalable plays an important role A proposal
for a quantum computer must at least in
principle be preparable or manufacturable in
large numbers of qubits, even if current
fundamental experiments are performed with Only
few qubits.

• A scalable physical system with well charactrized
  qubits
• The ability to initialize the state of the qubits
  to a simple fixed state
• Long relevant decoherence times, much longer than
  the gate operation time
• A universal set of quantum gates.
• A qubit-specific measurment capability.

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DiVincenzo Criteria

• A scalable physical system with well charactrized
  qubits
• The ability to initialize the state of the qubits
  to a simple fixed state
• Long relevant decoherence times, much longer than
  the gate operation time
• A universal set of quantum gates.
• A qubit-specific measurment capability.

every computation needs to be started in an
initially known state such as 000 . . .0gt.
Having also a fast initialization mechanism
at hand is crucial for quantum error correction
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DiVincenzo Criteria
Quantum states in contact with the outside world
ultimately evolve into a fully mixed states. By
encoding information not directly into single
qubits, but rather into logical qubits
consisting of several single qubits, a certain
amount of errors due to decoherence and
imperfect gates maybe corrected, depending on
what kind of code is used. There is however
still a limit on how faulty elementary gates are
allowed to be The accuracy threshold theorem
states that error correction is possible if the
error probability per gate is smaller than A
certain threshold. The threshold value depends on
the error models Studied and on the details of
the codes considered. Typical values are in the
range of to 10(-5) to 10(-3) , implying that
decoherence times must be a thousand to a
hundred thousand times longer than gate
operation times.

• A scalable physical system with well charactrized
  qubits
• The ability to initialize the state of the qubits
  to a simple fixed state
• Long relevant decoherence times, much longer than
  the gate operation time
• A universal set of quantum gates.
• A qubit-specific measurment capability.

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DiVincenzo Criteria
The generic quantum computing is possible in the
standard model if certain one- and two-qubit
gates are available. The single qubit gates may
be either implemented directly, or can be
approximated to arbitrary precision using a
finite set of gates. The only necessary
two-qubit gate is the controlled-not gate.

• A scalable physical system with well charactrized
  qubits
• The ability to initialize the state of the qubits
  to a simple fixed state
• Long relevant decoherence times, much longer than
  the gate operation time
• A universal set of quantum gates.
• A qubit-specific measurment capability.

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DiVincenzo Criteria
Measuring qubits without disturbing the rest of
the quantum computer is required in the
verification steps of quantum error correction
and, not remarkably, in order to reveal the
outcome of a computation. A meas- -urement is
said to have 100 quantum efficiency if it
yields, performed on a state
the
outcome 0 with probability p and 1 with
probability (1 - p) independent of a, the states
of neighboring qubits, or any other parameters
of the system. Real measure- -ments cannot have
perfect quantum efficiency.

• A scalable physical system with well charactrized
  qubits
• The ability to initialize the state of the qubits
  to a simple fixed state
• Long relevant decoherence times, much longer than
  the gate operation time
• A universal set of quantum gates.
• A qubit-specific measurment capability.

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Quantum dot
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QC with quantum dots
the physical system representing a qubit is
given by the localized spin state of one
electron, and the computational basis states
and are identified with the two spin
states and , respectively.
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QC with quantum dots
Scalability is due to the availability of local
gating. Gating operations are realized through
the exchange coupling, which can be tuned locally
with exponential precision.
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QC with quantum dots
In spin qubits, initialization could be achieved
by either forcing the spins to align with a
strong externally applied magnetic field, or by
performing a measurement on the dot followed by a
subsequent rotation of the state depending on the
measurement outcome.
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QC with quantum dots
Initialization of the quantum computer could be
realized at low temperature T by applying an
external magnetic field B satisfying gµB kT ,
where g is the g-factor, µ is Bohrs magneton,
and k is the Boltzmann constant. After a
sufficiently long time, virtually all spins will
have equilibrated to their thermodynamic ground
state 0gt ?gt.
Experiments are usually performed in dilution
Refrigerators with base temperature around 20
mK, which is smaller than typical
Zeeman splittings ( 300 mK at B 1 T and using
the bulk value g -0.44). The initialization
time is of the order of a few relaxation times,
which in GaAs dots have been reported to be as
high as 1 s.
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QC with quantum dots
In quantum dots where electron spins are used as
qubits, the most important mechanisms of
decoherence are the spin-orbit and the hyperfine
interaction
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QC with quantum dots

• Dresselhaus contribution
• Rashba contribution
• Hyperfine interaction

An electron moving through a solid experiences
electric fields, from charged atoms in the
lattice. This electric fields lead to a net
contribution to the spin- orbit interaction. This
effect is known as the Dresselhaus contribution
to the spin-orbit interaction and its
Hamiltonian for 2DEG with strong
confinement along growth direction (001) reads
Where depends on the material properties
and on
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QC with quantum dots
Dresselhaus contribution Rashba
contribution Hyperfine interaction
Electric fields associated with asymmetric
confining potentials also give rise to a
spin-orbit interaction (SIA, or structural
inversion asymmetry). The spin-orbit
contribution from SIA is known as the Rashba
term. Assuming that the confining electric field
is along the z axis, the Hamiltonian for 2DEG of
the Rashba contribution reads Where
depends on the material specific and on the
confining potential
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QC with quantum dots
Dresselhaus contribution Rashba
contribution Hyperfine interaction
The spin of an electron in an atom can interact
with the spin of its atomic nucleus through
the hyperfine coupling. An electron spin in a
quantum dot, in contrast, may interact with many
nuclear spins in the host material. The
Hamiltonian for the Fermi Contact hyperfine
interaction is then given by Where and
are the spin operator for nucleus k and the
electron spin Respectively
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Single spin rotations may be achieved by dragging
electrons down (by changing back gate voltages)
to a region where the Zeeman splitting in the
presence of the external static magnetic field
changes due to magnetization or an
inhomogeneous g-factor present in that layer. A
resonant magnetic ac pulse can then be used to
rotate the spin under consideration, while
leaving all other qubits unaffected due to the
off-resonant Zeeman splitting (ESR).
All-electrical single spin manipulation may be
realized in the presence of spin-orbit
interaction by applying ac electric pulses
directly via the gates (EDSR).
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QC with quantum dots
TWO-QUBIT GATES
The interaction of the two spins may be described
in terms of the isotropic Heisenberg Hamiltonian
Then the corresponding unitary evolution of the
state of the double dot is given
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QC with quantum dots
For the constant intraction and
time s.t
performs the so-called square-root of swap
denoted by This gate together with
single-qubit rotations about a fixed (say, the
z-) axis can be used to synthesize the cnot
operation as
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QC with quantum dots
Readout of electron spin states. Several methods
are available for reading out the spin state of
single and double quantum dots and all of them
rely on the mechanism of spin-to-charge
conversion.
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References - D.Loss, D.P. DiVincenzo Phys. Rev.
A 57, 120-126 (1998) - R. Hanson, L. P.
Kouwenhoven, J. R. Petta, S. Tarucha, L. M. K.
Vandersypen, Rev. Mod. Phys, 79, 1217 (2007)